New Lower Bound for the Optimal Ball Packing Density in Hyperbolic 4-Space

New horoball packing density lower bound in hyperbolic 5-space

“…Upper bounds for the packing density were published by Kellerhals [9] using the simplicial density function d n (∞). This bound is strict for n = 3, Table 1 summarizes our main results where ∆ is the gap between the upper and lower bounds at the time of writing, dimensions 3 ≤ n ≤ 5 were considered in previous work [10][11][12], while 6 ≤ n ≤ 9 are the subject of this paper.…”

“…However these ball packing configurations are only locally optimal and cannot be extended to the entirety of the ambeint space H n . In [11] we found seven horoball packings of Coxeter simplex tilings in H 4 that yield densities of The second-named author has several additional results on globally and locally optimal ball packings in H n , S n , and the eight Thurston geomerties arising from Thurston's geometrization conjecture [18][19][20]23]. Finally, in [12] we constructed the densest known ball packing in H 5 with a density of 5 7ζ(3) .…”

Decomposition method related to saturated hyperball packings

“…The plane α k 1 ...km and the hyperball H h s (P ) can be generated by rotation of φ and η about the common perpendicular σ; therefore, they are disjoint. Figure 2: The plane κ and its intersections with D(P ) and H h s (P ) 8. We have seen in steps 3, 4, 5 and 6 that the number of the outer vertices A k 1 ...km of any polyhedron obtained after the cutting process is less than the original one, and we have proven in step 7 that the original hyperballs form packings in the new polyhedra D 1 (P ) and D 2 (P ), as well.…”

“…3. What are the optimal horoball packing and covering configurations and what are their densities allowing horoballs in different types (n ≥ 4) (see [3,7,8])?…”

Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 3-space

“…In hyperbolic spaces H n (n ≥ 3) the problems of the densest horoball and hyperball packings have not been settled yet, in general (see e.g. [9], [19], [20]). Moreover, the optimal sphere packing problem can be extended to the other homogeneous Thurston geometries, e.g.…”